The Copernican Universe
Taking the Measure of the Solar System
While Copernicus' heliocentric model included a number of flaws and was no more accurate at predicting planetary positions than the Ptolemaic model it was designed to replace, the Copernican hypothesis did have the advantage of providing a means for calculating the distances to the planets. The geocentric model of Ptolemy was a useful mathematical fiction that was not considered to be the true representation of the heavens, later Arabic astronomers took the revered Greeks literally and interpreted the universe to be quite small, indeed.
With the advent of the heliocentric model, however, Copernicus demonstated it was possible to determine the distances between the Sun and the other planets with nothing more than keen observation and the geometry of Euclid. It is also important to understand a few basic definitions:
- Sidereal Period: The time required to complete one orbit around the Sun. As seen from Earth, the planet will complete one revolution around the celestial sphere.
- Synodic Period: The time between two successive identical alignments of the Earth, Sun and heavenly body. An example most people would identify with is the time between two Full Moons.
Inferior Planets
Inferior planets—those that orbit the Sun interior to Earth's orbit—have
four alignments with Earth that are of particular interest to observers,
either for what we can see or cannot see. The figure below shows those four
alignments. In all cases it is the alignment of the inferior planet with
both the Sun and Earth that is relevant, but for simplicity the position
of Earth is unchanged in the diagram.
At the moment of greatest elongation, an inferior planet is at its greatest angular separation from the Sun and appears to move radially towards/away from Earth. A triangle with the Sun, Earth and inferior planet at each vertex will contain a right (90°) angle at the vertex where the inferior planet is situation.
Using the law of sines, the Sun-Planet distance Y can be determined using the equation
Y = 1 A.U. • sinβ
where β is the angle of greatest elongation. Based on the geometry of the orbits, the most favorable time to observe an inferior planet is at greatest elongation—as a "morning star" at greatest western elongation or as an "evening star" at greatest eastern elongation.
Superior Planets
There are also four significant orientations of superior planets
and two of them lead to a determination of the planetary distance.
The calculation is somewhat more involved, however.
The most favorable time to observe a superior planet is when it is at opposition. At that time the planet is placed 180° from the Sun and visible all night long. It is also at its closest approach to Earth and therefore brightest and largest in our sky.
At quadrature the superior planet is 90° from the Sun in our sky and there is a right angle at the vertex centered on Earth. The angle of the vertex centered on the superior planet can be determined from the time between opposition and quadrature and the synodic frequency.
In other words,
β = 90° - the angle whose derivative is the synodic frequency
and
Y = 1 A.U. / sin β .